Green function for the upper-half space $\mathbb{R}^{n+1}_+$

Question

Given $\mathbb{R}^{n+1}_+=\{X=(x,t): x \in \mathbb{R}^n , t>0 \}$ as domain, what is the explicit formula for the Green function $G(X,Y)$ for the Laplacian on $\mathbb{R}^{n+1}_+$? I know that fundamental fundamental solution on $\mathbb{R}^{n+1}_+$; but want to have an explicit formula for the Green function.

Answer 1

It is done via reflection, also known as method of images. Set $\mathbb{R}^{n+1}_-=\{(u,t) \mid u \in \mathbb{R}^n , t <0 \}$ Consider the mapping $$ R: \mathbb{R}^{n+1}_+ \to \mathbb{R}^{n+1}_-, (u,t) \mapsto (u,-t). $$ Then Greens function for $x,y \in \mathbb{R}^{n+1}_+$ is given by $$ G(x,y)=\Phi(y-x)- \Phi(y-Rx), $$ where $\Phi$ is the fundamental solution on $\mathbb{R}^{n+1}$. This then becomes $$ G(x,y)= -\frac{1}{(n+1) C(n+1) } \bigg( \frac{t_y-t_x}{|y-x|^n}+\frac{t_y+t_x}{|y-Rx|} \bigg), $$ where $C(n+1)$ is the volume of the unit ball in $\mathbb{R}^{n+1}$ and $t_y$ and $t_x$ are the last component of $y$ and $x$, respectively.